Fractals, topology, complex arithmetic and fascinating computer graphics.


 Cleopatra Katrina Freeman
 1 years ago
 Views:
Transcription
1 Chapter 13 Mandelbrot Set Fractals, topology, complex arithmetic and fascinating computer graphics. Benoit Mandelbrot was a Polish/French/American mathematician who has spent most of his career at the IBM Watson Research Center in Yorktown Heights, N.Y. He coined the term fractal and published a very influential book, The Fractal Geometry of Nature, in An image of the now famous Mandelbrot set appeared on the cover of Scientific American in This was about the time that computer graphical displays were becoming widely available. Since then, the Mandelbrot set has stimulated deep research topics in mathematics and has also been the basis for an uncountable number of graphics projects, hardware demos, and Web pages. To get in the mood for the Mandelbrot set, consider the region in the complex plane of trajectories generated by repeated squaring, z k+1 = z 2 k, k = 0, 1,... For which initial values z 0 does this sequence remain bounded as k? It is easy to see that this set is simply the unit disc, z 0 <= 1, shown in figure If z 0 <= 1, the sequence z k remains bounded. But if z 0 > 1, the sequence is unbounded. The boundary of the unit disc is the unit circle, z 0 = 1. There is nothing very difficult or exciting here. The definition is the Mandelbrot set is only slightly more complicated. It involves repeatedly adding in the initial point. The Mandelbrot set is the region in the complex plane consisting of the values z 0 for which the trajectories defined by z k+1 = z 2 k + z 0, k = 0, 1,... remain bounded at k. That s it. That s the entire definition. It s amazing that such a simple definition can produce such fascinating complexity. Copyright c 2011 Cleve Moler Matlab R is a registered trademark of MathWorks, Inc. TM October 2,
2 2 Chapter 13. Mandelbrot Set Figure The unit disc is shown in red. The boundary is simply the unit circle. There is no intricate fringe Figure The Mandelbrot set is shown in red. The fringe just outside the set, shown in black, is a region of rich structure.
3 3 Figure Two trajectories. z0 = i generates a cycle of length four, while nearby z0 = i generates an unbounded trajectory. Figure 13.2 shows the overall geometry of the Mandelbrot set. However, this view does not have the resolution to show the richly detailed structure of the fringe just outside the boundary of the set. In fact, the set has tiny filaments reaching into the fringe region, even though the fringe appears to be solid black in the figure. It has recently been proved that the Mandelbrot set is mathematically connected, but the connected region is sometimes so thin that we cannot resolve it on a graphics screen or even compute it in a reasonable amount of time. To see how the definition works, enter z0 = i z = 0 into Matlab. Then use the uparrow key to repeatedly execute the statement z = z^2 + z0 The first few lines of output are i i i i i... The values eventually settle into a cycle i i i i
4 4 Chapter 13. Mandelbrot Set i... This cycle repeats forever. The trajectory remains bounded. This tells us that the starting value value, z0 = i, is in the Mandelbrot set. The same cycle is shown in the left half of figure On the other hand, start with z0 = i z = 0 and repeatedly execute the statement You will see z = z^2 + z i i i i i... Then, after 24 iterations, i i i e e+002i e e+004i The trajectory is blowing up rapidly. After a few more iterations, the floating point numbers overflow. So this z0 is not in the Mandelbrot set. The same unbounded trajectory is shown in the right half of figure We see that the first value, z0 = i, is in the Mandelbrot set, while the second value, z0 = i, which is nearby, is not. The algorithm doesn t have to wait until z reachs floating point overflow. As soon as z satisfies abs(z) >= 2 subsequent iterations will essential square the value of z and it will behave like 2 2k. Try it yourself. Put these statements on one line. z0 =... z = 0; while abs(z) < 2 z = z^2+z0; disp(z), end
5 5 Use the up arrow and backspace keys to retrieve the statement and change z0 to different values near i. If you have to hit <ctrl>c to break out of an infinite loop, then z0 is in the Mandelbrot set. If the while condition is eventually false and the loop terminates without your help, then z0 is not in the set. The number of iterations required for z to escape the disc of radius 2 provides the basis for showing the detail in the fringe. Let s add an iteration counter to the loop. A quantity we call depth specifies the maximum interation count and thereby determines both the level of detail and the overall computation time. Typical values of depth are several hundred or a few thousand. z0 =... z = 0; k = 0; while abs(z) < 2 && k < depth z = z^2+z0; k = k + 1; end The maximum value of k is depth. If the value of k is less than depth, then z0 is outside the set. Large values of k indicate that z0 is in the fringe, close to the boundary. If k reaches depth then z0 is declared to be inside the Mandelbrot set. Here is a small table of iteration counts s z0 ranges over complex values near i. We have set depth = We see that about half of the values are less than depth; they correspond to points outside of the Mandelbrot set, in the fringe near the boundary. The other half of the values are equal to depth, corresponding to points that are regarded as in the set. If we were to redo the computation with a larger value of depth, the entries that are less than 512 in this table would not change, but some of the entries that are now capped at 512 might increase. The iteration counts can be used as indices into an RGB color map of size depthby3. The first row of this map specifies the color assigned to any points on the z0 grid that lie outside the disc of radius 2. The next few rows provide colors for the points on the z0 grid that generate trajectories that escape quickly. The last row of the map is the color of the points that survive depth iterations and so are in the set.
6 6 Chapter 13. Mandelbrot Set The map used in figure 13.2 emphasizes the set itself and its boundary. The map has 12 rows of white at the beginning, one row of dark red at the end, and black in between. Images that emphasize the structure in the fringe are achieved when the color map varies cyclicly over a few dozen colors. One of the exercises asks you to experiment with color maps. Figure Improving resolution. Array operations. Our script mandelbrot_recap shows how Matlab array arithmetic operates a grid of complex numbers simultaneously and accumulates an array of iteration counters, producing images like those in figure 13.4 The code begins by defining the region in the complex plane to be sampled. A step size of 0.05 gives the coarse resolution shown on the right in the figure. x = 0: 0.05: 0.80; y = x ; The next section of code uses an elegant, but tricky, bit of Matlab indexing known as Tony s Trick. The quantities x and y are onedimensional real arrays of length n, one a column vector and the other a row vector. We want to create a twodimensional nbyn array with elements formed from all possible sums of elements from x and y. z k,j = x k + y j i, i = 1, k, j = 1,..., n This can be done by generating a vector e of length n with all elements equal to one. Then the quantity x(e,:) is a twodimensional array formed by using x, which is the same as x(1,:), n times. Similarly, y(:,e) is a twodimensional arrray containing n copies of the column vector y.
7 7 n = length(x); e = ones(n,1); z0 = x(e,:) + i*y(:,e); If you find it hard to remember Tony s indexing trick, the function meshgrid does the same thing in two steps. [X,Y] = meshgrid(x,y); z0 = X + i*y; Now initialize two more arrays, one for the complex iterates and one for the counts. z = zeros(n,n); c = zeros(n,n); Here is the Mandelbrot iteration repeated depth times. With each iteration we also keep track of the iterates that are still within the circle of radius 2. depth = 32; for k = 1:depth z = z.^2 + z0; c(abs(z) < 2) = k; end We are now finished with z. The actual values of z are not important, only the counts are needed to create the image. Our grid is small enough that we can actually print out the counts c. c The results are c =
8 8 Chapter 13. Mandelbrot Set We see that points in the upper left of the grid, with fairly small initial z0 values, have survived 32 iterations without going outside the circle of radius two, while points in the lower right, with fairly large initial values, have lasted only one or two iterations. The interesting grid points are in between, they are on the fringe. Now comes the final step, making the plot. The image command does the job, even though this is not an image in the usual sense. The count values in c are used as indices into a 32by3 array of RGB color values. In this example, the jet colormap is reversed to give dark red as its first value, pass through shades of green and yellow, and finish with dark blue as its 32nd and final value. image(c) axis image colormap(flipud(jet(depth))) Exercises ask you to increase the resolution by decreasing the step size, thereby producing the other half of figure 13.4, to investigate the effect of changing depth, and to invesigate other color maps. Mandelbrot GUI The exm toolbox function mandelbrot is your starting point for exploration of the Mandelbrot set. With no arguments, the statement mandelbrot provides thumbnail icons of the twelve regions featured in this chapter. The statement mandelbrot(r) with r between 1 and 12 starts with the rth region. The statement mandelbrot(center,width,grid,depth,cmapindx) explores the Mandelbrot set in a square region of the complex plane with the specified center and width, using a gridbygrid grid, an iteration limit of depth, and the color map number cmapindx. The default values of the parameters are center = i width = 3 grid = 512 depth = 256 cmapindx = 1 In other words, mandelbrot(0.5+0i, 3, 512, 256, 1) generates figure 13.2, but with the jets color map. Changing the last argument from 1 to 6 generates the actual figure 13.2 with the fringe color map. On my laptop, these computations each take about half a second. A simple estimate of the execution time is proportional to
9 9 grid^2 * depth So the statement could take mandelbrot(0.5+0i, 3, 2048, 1024, 1) (2048/512) 2 (1024/256) = 64 times as long as the default. However, this is an overestimate and the actual execution time is about 11 seconds. Most of the computational time required to compute the Mandelbrot set is spent updating two arrays z and kz by repeatedly executing the step z = z.^2 + z0; j = (abs(z) < 2); kz(j) = d; This computation can be carried out faster by writing a function mandelbrot_step in C and creating as a Matlab executable or cmex file. Different machines and operating systems require different versions of a mex file, so you should see files with names like mandelbrot_step.mexw32 and mandelbrot_step.glnx64 in the exm toolbox. The mandelbrot gui turns on the Matlab zoom feature. The mouse pointer becomes a small magnifying glass. You can click and release on a point to zoom by a factor of two, or you can click and drag to delineate a new region. The mandelbrot gui provides several uicontrols. Try these as you read along. The listbox at the bottom of the gui allows you to select any of the predefined regions shown in the figures in this chapter. depth. Increase the depth by a factor of 3/2 or 4/3. grid. Refine the grid by a factor of 3/2 or 4/3. The depth and grid size are always a power of two or three times a power of two. Two clicks on the depth or grid button doubles the parameter. color. Cycle through several color maps. jets and hots are cyclic repetitions of short copies of the basic Matlab jet and hot color maps. cmyk cycles through eight basic colors, blue, green, red, cyan, magenta, yellow, gray, and black. fringe is a noncyclic map used for images like figure exit. Close the gui. The Mandelbrot set is self similar. Small regions in the fringe reveal features that are similar to the original set. Figure 13.6, which we have dubbed Mandelbrot
10 10 Chapter 13. Mandelbrot Set Figure The figures in this chapter, and the predefined regions in our mandelbrot program, show these regions in the fringe just outside the Mandelbrot set. Junior, is one example. Figure 13.7, which we call the Plaza, uses our flag colormap to reveal fine detail in red, white and blue. The portion of the boundary of the Mandelbrot set between the two large, nearly circular central regions is known as The Valley of the Seahorses. Figure 13.8 shows the result of zooming in on the peninsula between the two nearly circular regions of the set. The figure can be generated directly with the command mandelbrot( i,0.1,1024,512) We decided to name the image in figure 13.9 the West Wing because it resembles the Xwing fighter that Luke Skywalker flies in Star Wars and because it is located near the leftmost, or far western, portion of the set. The magnification factor is a relatively modest 10 4, so depth does not need to be very large. The command to generate the West Wing is mandelbrot( i,1.5e4,1024,160,1) One of the best known examples of self similarity, the Buzzsaw, is shown in figure It can be generated with mandelbrot( i,...
11 11 4.0e11,1024,2048,2) Taking width = 4.0e11 corresponds to a magnification factor of almost To appreciate the size of this factor, if the original Mandelbrot set fills the screen on your computer, the Buzzsaw is smaller than the individual transistors in your machine s microprocessor. We call figure Nebula because it reminds us of interstellar dust. It is generated by mandelbrot( i,4.88e5,1024,2048,3) The next three images are obtained by carefully zooming on one location. We call them the Vortex, the Microbug, and the Nucleus. mandelbrot( i, e12,1024,2048,2) mandelbrot( i, e13,1024,2048,2) mandelbrot( i, e14,1024,2048,2) The most intricate and colorful image among our examples is figure 13.16, the Geode. It involves a fine grid and a large value of depth and consequently requires a few minutes to compute. mandelbrot( i,6.0e10,2048,4096,1) set. These examples are just a tiny sampling of the structure of the Mandelbrot Further Reading We highly recommend a real time fractal zoomer called XaoS, developed by Thomas Marsh, Jan Hubicka and Zoltan Kovacs, assisted by an international group of volunteers. See If you are expert at using your Web browser and possibly downloading an obscure video codec, take a look at the Wikipedia video Image:Fractalzoom103Mandelbrot_Buzzsaw.ogg It s terrific to watch, but it may be a lot of trouble to get working. Recap %% Mandelbrot Chapter Recap % This is an executable program that illustrates the statements
12 12 Chapter 13. Mandelbrot Set % introduced in the Mandelbrot Chapter of "Experiments in MATLAB". % You can access it with % % mandelbrot_recap % edit mandelbrot_recap % publish mandelbrot_recap % % Related EXM programs % % mandelbrot %% Define the region. x = 0: 0.05: 0.8; y = x ; %% Create the twodimensional complex grid using Tony s indexing trick. n = length(x); e = ones(n,1); z0 = x(e,:) + i*y(:,e); %% Or, do the same thing with meshgrid. [X,Y] = meshgrid(x,y); z0 = X + i*y; %% Initialize the iterates and counts arrays. z = zeros(n,n); c = zeros(n,n); %% Here is the Mandelbrot iteration. depth = 32; for k = 1:depth z = z.^3 + z0; c(abs(z) < 2) = k; end %% Create an image from the counts. c image(c) axis image %% Colors colormap(flipud(jet(depth)))
13 13 Exercises 13.1 Explore. Use the mandelbrot gui to find some interesting regions that, as far as you know, have never been seen before. Give them your own names depth. Modify mandelbrot_recap to reproduce our table of iteration counts for x =.205:.005:.245 and y = .520:.005: First, use depth = 512. Then use larger values of depth and see which table entries change Resolution. Reproduce the image in the right half of figure Big picture. Modify mandelbrot_recap to display the entire Mandelbrot set Color maps. Investigate color maps. Use mandelbrot_recap with a smaller step size and a large value of depth to produce an image. Find how mandelbrot computes the cyclic color maps called jets, hots and sepia. Then use those maps on your image pth power. In either mandelbrot_recap or the mandelbrot gui, change the power in the Mandelbrot iteration to z k+1 = z p k + z 0 for some fixed p 2. If you want to try programming Handle Graphics, add a button to mandelbrot that lets you set p Too much magnification. When the width of the region gets to be smaller than about 10 15, our mandelbrot gui does not work very well. Why? 13.8 Spin the color map. This might not work very well on your computer because it depends on what kind of graphics hardware you have. When you have an interesting region plotted in the figure window, bring up the command window, resize it so that it does not cover the figure window, and enter the command spinmap(10) I won t try to describe what happens you have to see it for yourself. The effect is most dramatic with the seahorses2 region. Enter help spinmap for more details.
14 14 Chapter 13. Mandelbrot Set Figure Region #2, Mandelbrot Junior. The fringe around the Mandelbrot set in self similar. Small versions of the set appear at all levels of magnification. Figure Region #3, Plaza, with the flag color map.
15 15 Figure Region #4. Valley of the Seahorses. Figure Region #5. Our West Wing is located just off the real axis in the thin far western portion of the set, near real(z) =
16 16 Chapter 13. Mandelbrot Set Figure Region #6. Dueling Dragons. Figure Region #7. The Buzzsaw requires a magnification factor of to reveal a tiny copy of the Mandelbrot set.
17 17 Figure Region #8. Nebula. Interstellar dust. Figure Region #9. A vortex, not far from the West Wing. Zoom in on one of the circular microbugs near the left edge.
18 18 Chapter 13. Mandelbrot Set Figure Region #10. A magnification factor reveals a microbug within the vortex. Figure Region #11. The nucleus of the microbug.
19 Figure Region #12. Geode. This colorful image requires a by2048 grid and depth =
mouse (or the option key on Macintosh) and move the mouse. You should see that you are able to zoom into and out of the scene.
A Ball in a Box 1 1 Overview VPython is a programming language that is easy to learn and is well suited to creating 3D interactive models of physical systems. VPython has three components that you will
More informationUser Tutorial on Changing Frame Size, Window Size, and Screen Resolution for The Original Version of The CancerRates.Info/NJ Application
User Tutorial on Changing Frame Size, Window Size, and Screen Resolution for The Original Version of The CancerRates.Info/NJ Application Introduction The original version of CancerRates.Info/NJ, like
More informationGetting Started in Tinkercad
Getting Started in Tinkercad By Bonnie Roskes, 3DVinci Tinkercad is a fun, easy to use, webbased 3D design application. You don t need any design experience  Tinkercad can be used by anyone. In fact,
More informationGetting Started in Microsoft Word Click on the Start menu > All Programs > Microsoft Office and select Microsoft Office Word 2007 from the list.
Microsoft Word Part 1 Office 2007 Getting Started in Microsoft Word 2007 Opening Word Double click the Word icon on the desktop. OR If there is no shortcut icon on the desktop: Click on the Start menu
More informationVisualization of 2D Domains
Visualization of 2D Domains This part of the visualization package is intended to supply a simple graphical interface for 2 dimensional finite element data structures. Furthermore, it is used as the low
More informationMICROSOFT OFFICE ACCESS 2007  NEW FEATURES
MICROSOFT OFFICE 2007 MICROSOFT OFFICE ACCESS 2007  NEW FEATURES Exploring Access Creating and Working with Tables Finding and Filtering Data Working with Queries and Recordsets Working with Forms Working
More informationAdobe Illustrator CS5 Part 1: Introduction to Illustrator
CALIFORNIA STATE UNIVERSITY, LOS ANGELES INFORMATION TECHNOLOGY SERVICES Adobe Illustrator CS5 Part 1: Introduction to Illustrator Summer 2011, Version 1.0 Table of Contents Introduction...2 Downloading
More informationUsing Microsoft Picture Manager
Using Microsoft Picture Manager Storing Your Photos It is suggested that a county store all photos for use in the County CMS program in the same folder for easy access. For the County CMS Web Project it
More informationIf you know exactly how you want your business forms to look and don t mind detail
Advanced Form Customization APPENDIX E If you know exactly how you want your business forms to look and don t mind detail work, you can customize QuickBooks forms however you want. With QuickBooks Layout
More informationHermes.Net IVR Designer Page 2 36
Hermes.Net IVR Designer Page 2 36 Summary 1. Introduction 4 1.1 IVR Features 4 2. The interface 5 2.1 Description of the Interface 6 2.1.1 Menus. Provides 6 2.1.2 Commands for IVR editions. 6 2.1.3 Commands
More informationThe following is an overview of lessons included in the tutorial.
Chapter 2 Tutorial Tutorial Introduction This tutorial is designed to introduce you to some of Surfer's basic features. After you have completed the tutorial, you should be able to begin creating your
More informationIntroduction to Microsoft Excel 2007
Introduction to Microsoft Excel 2007 Class learning objectives 1. What is Excel? Spreadsheet uses & samples Touring the Excel window Learning important definitions Navigating around the workbook 2. The
More informationSMART Notebook 10 User s Guide. Linux Operating Systems
SMART Notebook 10 User s Guide Linux Operating Systems Product Registration If you register your SMART product, we ll notify you of new features and software upgrades. Register online at www.smarttech.com/registration.
More informationCS1112 Spring 2014 Project 4. Objectives. 3 Pixelation for Identity Protection. due Thursday, 3/27, at 11pm
CS1112 Spring 2014 Project 4 due Thursday, 3/27, at 11pm You must work either on your own or with one partner. If you work with a partner you must first register as a group in CMS and then submit your
More informationIntro to PowerPoint 2010
Instructional Center for Educational Technologies Intro to PowerPoint 2010 PowerPoint is a multimedia tool that serves as a visual aid to your content and delivery. When used effectively, PowerPoint can
More informationIntellect Platform  The Workflow Engine Basic HelpDesk Troubleticket System  A102
Intellect Platform  The Workflow Engine Basic HelpDesk Troubleticket System  A102 Interneer, Inc. Updated on 2/22/2012 Created by Erika Keresztyen Fahey 2 Workflow  A102  Basic HelpDesk Ticketing System
More informationPhotoshop Image Editing
Photoshop Image Editing Opening a file: File Menu > Open Photoshop Workspace A: Menus B: Application Bar view options, etc. C: Options bar controls specific to the tool you are using at the time. D:
More informationVanderbilt University School of Nursing. Running Scopia Videoconferencing from Windows
Vanderbilt University School of Nursing Running Scopia Videoconferencing from Windows gordonjs 3/4/2011 Table of Contents Contents Installing the Software... 3 Configuring your Audio and Video... 7 Entering
More informationm ac romed ia Fl a s h Curriculum Guide
m ac romed ia Fl a s h Curriculum Guide 1997 1998 Macromedia, Inc. All rights reserved. Macromedia, the Macromedia logo, Dreamweaver, Director, Fireworks, Flash, Fontographer, FreeHand, and Xtra are trademarks
More informationSpreadsheet. Parts of a Spreadsheet. Entry Bar
Spreadsheet Parts of a Spreadsheet 1. Open the AppleWorks program. Select spreadsheet. 2. Explore the spreadsheet setup for a while. Active Cell Address Entry Bar Column Headings Row Headings Active Cell
More informationWhat s New V 11. Preferences: Parameters: Layout/ Modifications: Reverse mouse scroll wheel zoom direction
What s New V 11 Preferences: Reverse mouse scroll wheel zoom direction Assign mouse scroll wheel Middle Button as Fine tune Pricing Method (Manufacturing/Design) Display Display Long Name Parameters:
More informationPA Payroll Exercise for Intermediate Excel
PA Payroll Exercise for Intermediate Excel Follow the directions below to create a payroll exercise. Read through each individual direction before performing it, like you are following recipe instructions.
More informationOPERATION MANUAL. MV410RGB Layout Editor. Version 2.1 higher
OPERATION MANUAL MV410RGB Layout Editor Version 2.1 higher Table of Contents 1. Setup... 1 11. Overview... 1 12. System Requirements... 1 13. Operation Flow... 1 14. Installing MV410RGB Layout
More informationIntroduction to MATLAB
Introduction to MATLAB Matlab is a program that allows you to carry out computations in a straightforward manner, removing much of the tedium involved in programming. It is extremely useful for creating
More informationIntro to Excel spreadsheets
Intro to Excel spreadsheets What are the objectives of this document? The objectives of document are: 1. Familiarize you with what a spreadsheet is, how it works, and what its capabilities are; 2. Using
More informationExcel 2007: Basics Learning Guide
Excel 2007: Basics Learning Guide Exploring Excel At first glance, the new Excel 2007 interface may seem a bit unsettling, with fat bands called Ribbons replacing cascading text menus and task bars. This
More informationHow to Make Animated Graphs. 2. Select the cells highlighted in yellow, and click on the chart wizard. Create an xy scatter chart.
How to Make Animated Graphs This tutorial will walk you through the steps of creating a simple animated graph in Excel. The end product is an animated scatter plot (similar to Gapminder World), but the
More informationPlotting: Customizing the Graph
Plotting: Customizing the Graph Data Plots: General Tips Making a Data Plot Active Within a graph layer, only one data plot can be active. A data plot must be set active before you can use the Data Selector
More information2013 Getting Started Guide
2013 Getting Started Guide The contents of this guide and accompanying exercises were originally created by Nemetschek Vectorworks, Inc. Vectorworks Fundamentals Getting Started Guide Created using: Vectorworks
More informationUnderstand the Sketcher workbench of CATIA V5.
Chapter 1 Drawing Sketches in Learning Objectives the Sketcher WorkbenchI After completing this chapter you will be able to: Understand the Sketcher workbench of CATIA V5. Start a new file in the Part
More informationDrawing a histogram using Excel
Drawing a histogram using Excel STEP 1: Examine the data to decide how many class intervals you need and what the class boundaries should be. (In an assignment you may be told what class boundaries to
More informationAcrobat PDF Forms  Part 2
Acrobat PDF Forms  Part 2 PDF Form Fields In this lesson, you will be given a file named Information Request Form that can be used in either Word 2003 or Word 2007. This lesson will guide you through
More informationBlender Notes. Introduction to Digital Modelling and Animation in Design Blender Tutorial  week 9 The Game Engine
Blender Notes Introduction to Digital Modelling and Animation in Design Blender Tutorial  week 9 The Game Engine The Blender Game Engine This week we will have an introduction to the Game Engine build
More informationEpson Brightlink Interactive Board and Pen Training. Step One: Install the Brightlink Easy Interactive Driver
California State University, Fullerton Campus Information Technology Division Documentation and Training Services Handout Epson Brightlink Interactive Board and Pen Training Downloading Brightlink Drivers
More informationIn this session, we will explain some of the basics of word processing. 1. Start Microsoft Word 11. Edit the Document cut & move
WORD PROCESSING In this session, we will explain some of the basics of word processing. The following are the outlines: 1. Start Microsoft Word 11. Edit the Document cut & move 2. Describe the Word Screen
More informationGraphic Design Basics Tutorial
Graphic Design Basics Tutorial This tutorial will guide you through the basic tasks of designing graphics with Macromedia Fireworks MX 2004. You ll get handson experience using the industry s leading
More informationHow to blend projection images on a curved screen using the Warp Generator
How to blend projection images on a curved screen using the Warp Generator version 2.1 Introduction The guide describes how to blend two projection images accurately onto a curved screen using the Calibre
More informationOtis Photo Lab Inkjet Printing Demo
Otis Photo Lab Inkjet Printing Demo Otis Photography Lab Adam Ferriss Lab Manager aferriss@otis.edu 310.665.6971 Soft Proofing and Pre press Before you begin printing, it is a good idea to set the proof
More informationMARS STUDENT IMAGING PROJECT
MARS STUDENT IMAGING PROJECT Data Analysis Practice Guide Mars Education Program Arizona State University Data Analysis Practice Guide This set of activities is designed to help you organize data you collect
More informationBasic Excel Handbook
2 5 2 7 1 1 0 4 3 9 8 1 Basic Excel Handbook Version 3.6 May 6, 2008 Contents Contents... 1 Part I: Background Information...3 About This Handbook... 4 Excel Terminology... 5 Excel Terminology (cont.)...
More informationMicrosoft PowerPoint 2008
Microsoft PowerPoint 2008 Starting PowerPoint... 2 Creating Slides in Your Presentation... 3 Beginning with the Title Slide... 3 Inserting a New Slide... 3 Slide Layouts... 3 Adding an Image to a Slide...
More informationDigital Image Processing. Prof. P.K. Biswas. Department of Electronics & Electrical Communication Engineering
Digital Image Processing Prof. P.K. Biswas Department of Electronics & Electrical Communication Engineering Indian Institute of Technology, Kharagpur Lecture  27 Colour Image Processing II Hello, welcome
More informationMicrosoft Office 2010
Word Tutorial 4 Desktop Publishing and Mail Merge Microsoft Office 2010 Objectives Identify desktop publishing features Format text in columns Insert drop caps Insert symbols and special characters Create
More informationTutorial: 2D Pipe Junction Using Hexa Meshing
Tutorial: 2D Pipe Junction Using Hexa Meshing Introduction In this tutorial, you will generate a mesh for a twodimensional pipe junction, composed of two inlets and one outlet. After generating an initial
More informationPowerPoint 2007 Basics Website: http://etc.usf.edu/te/
Website: http://etc.usf.edu/te/ PowerPoint is the presentation program included in the Microsoft Office suite. With PowerPoint, you can create engaging presentations that can be presented in person, online,
More informationOperating Systems. and Windows
Operating Systems and Windows What is an Operating System? The most important program that runs on your computer. It manages all other programs on the machine. Every PC has to have one to run other applications
More informationCreating Interactive PDF Forms
Creating Interactive PDF Forms Using Adobe Acrobat X Pro Information Technology Services Outreach and Distance Learning Technologies Copyright 2012 KSU Department of Information Technology Services This
More informationWinter Main 2014 Spring MidMonth 2015. Online Practice Test Directions
Georgia Milestones Winter Main 2014 Spring MidMonth 2015 Online Practice Test Directions EndofCourse Online Practice Test Directions 2706773W TEST SECURITY Below is a list, although not inclusive,
More informationMassArt Studio Foundation: Visual Language Digital Media Cookbook, Fall 2013
INPUT OUTPUT 08 / IMAGE QUALITY & VIEWING In this section we will cover common image file formats you are likely to come across and examine image quality in terms of resolution and bit depth. We will cover
More informationTable of Contents TASK 1: DATA ANALYSIS TOOLPAK... 2 TASK 2: HISTOGRAMS... 5 TASK 3: ENTER MIDPOINT FORMULAS... 11
Table of Contents TASK 1: DATA ANALYSIS TOOLPAK... 2 TASK 2: HISTOGRAMS... 5 TASK 3: ENTER MIDPOINT FORMULAS... 11 TASK 4: ADD TOTAL LABEL AND FORMULA FOR FREQUENCY... 12 TASK 5: MODIFICATIONS TO THE HISTOGRAM...
More informationCreating a Newsletter with Microsoft Word
Creating a Newsletter with Microsoft Word Frank Schneemann In this assignment we are going to use Microsoft Word to create a newsletter that can be used in your classroom instruction. If you already know
More informationWorking with images in Word 2013
Working with images in Word 2013 Information Services Working with images in Word 2013 Pictures and illustrations can be an effective way of adding interest or getting your message across. This note covers
More informationNXT Generation Robotics
NXT Generation Robotics Introductory Worksheets School of Computing University of Kent Copyright c 2010 University of Kent NXT Generation Robotics These worksheets are intended to provide an introduction
More informationMET 306 Activity 6. Using Pro/MFG Milling Operations Creo 2.0. Machining a Mast Step
Using Pro/MFG Milling Operations Creo 2.0 Machining a Mast Step If the Trim option is grayed out when trimming the mill volume, Save (making sure the.asm file is going to the correct subdirectory), Exit
More informationSharing Files and Whiteboards
Your user role in a meeting determines your level of file sharing. The type of files you can share include documents, presentations, and videos. About Sharing Files, page 1 Changing Views in a File or
More informationLession: 2 Animation Tool: Synfig Card or Page based Icon and Event based Time based Pencil: Synfig Studio: Getting Started: Toolbox Canvas Panels
Lession: 2 Animation Tool: Synfig In previous chapter we learn Multimedia and basic building block of multimedia. To create a multimedia presentation using these building blocks we need application programs
More informationA Quick Start Guide to Using PowerPoint For Imagebased Presentations University of Chicago, Department of Art History Visual Resources Collection
A Quick Start Guide to Using PowerPoint For Imagebased Presentations University of Chicago, Department of Art History Visual Resources Collection Adapted from Using PowerPoint To Create an Art History
More informationDOING MORE WITH WORD: MICROSOFT OFFICE 2010
University of North Carolina at Chapel Hill Libraries Carrboro Cybrary Chapel Hill Public Library Durham County Public Library DOING MORE WITH WORD: MICROSOFT OFFICE 2010 GETTING STARTED PAGE 02 Prerequisites
More informationSMART Ink 1.5. Windows operating systems. Scan the following QR code to view the SMART Ink Help on your smart phone or other mobile device.
SMART Ink 1.5 Windows operating systems User s guide Scan the following QR code to view the SMART Ink Help on your smart phone or other mobile device. Trademark notice SMART Ink, SMART Notebook, SMART
More informationSMART Board Interactive Whiteboard Setup with USB Cable
SMART Board Interactive Whiteboard Setup with USB Cable The instructions below are for the SMART Board interactive whiteboard 500 series and apply to both desktop and laptop computers. Ready Light USB
More informationQuickstart for Desktop Version
Quickstart for Desktop Version What is GeoGebra? Dynamic Mathematics Software in one easytouse package For learning and teaching at all levels of education Joins interactive 2D and 3D geometry, algebra,
More informationLab 3. GIS Data Entry and Editing.
Lab 3. GIS Data Entry and Editing. The goal: To learn about the vector (arc/node) and raster data types entry and editing. Objective: Create vector and raster datasets and visualize them. Software for
More informationLaser Cutting from Adobe Illustrator CS6 on VLS6.60 (For Adobe Illustrator CS6) for CARDINAL & BLUE JAY If you re starting a new document
Laser Cutting from Adobe Illustrator CS6 on VLS6.60 (For Adobe Illustrator CS6) for CARDINAL & BLUE JAY If you re starting a new document 1. Open Adobe Illustrator CS6 and go to file then new (If your
More informationIntroduction. Page1of13
BookMap User Guide Introduction BookMap is an innovative analysis and trading application that delivers a consolidated visualization of past and present market depth. Using a configurable color scheme,
More informationSmartboard and Notebook 10 What s New
Smartboard and Notebook 10 What s New Smartboard Markers and Eraser (for use with all programs): You may use your finger as the mouse. Press down twice to double click Hold your finger down for a few seconds
More informationSMART Board Training Packet. Notebook Software 10.0
SMART Board Training Packet Notebook Software 10.0 Chris de Treville Chris.deTreville@avispl.com 6301 Benjamin Road Suite 101 Tampa, FL 33634 p: 813.884.7168 f: 813.882.9508 SMART Board Welcome Center
More informationAwesome PowerPoint Tricks for Effective Presentations
Awesome PowerPoint Tricks for Effective Presentations Visualization Identify all the individual elements that could be represented by a single object, icon, or picture Role Person Company X Y Z Graph X
More informationHow to download historical data for a list of securities from outside of Bloomberg using Bloomberg Excel API s History Wizard
How to download historical data for a list of securities from outside of Bloomberg using Bloomberg Excel API s History Wizard There are two primary methods to download historical data for a list of securities
More informationFiles Used in this Tutorial
Generate Point Clouds Tutorial This tutorial shows how to generate point clouds from IKONOS satellite stereo imagery. You will view the point clouds in the ENVI LiDAR Viewer. The estimated time to complete
More informationGUIDELINES FOR PREPARING POSTERS USING POWERPOINT PRESENTATION SOFTWARE
Society for the Teaching of Psychology (APA Division 2) OFFICE OF TEACHING RESOURCES IN PSYCHOLOGY (OTRP) Department of Psychology, Georgia Southern University, P. O. Box 8041, Statesboro, GA 304608041
More informationSpaceClaim Introduction Training Session. A SpaceClaim Support Document
SpaceClaim Introduction Training Session A SpaceClaim Support Document In this class we will walk through the basic tools used to create and modify models in SpaceClaim. Introduction We will focus on:
More informationChapter 9 Slide Shows
Impress Guide Chapter 9 Slide Shows Transitions, animations, and more Copyright This document is Copyright 2007 2013 by its contributors as listed below. You may distribute it and/or modify it under the
More informationIntermediate Python Tutorial
Intermediate Python Tutorial Python is free software and is an easy to learn powerful programming language. It has efficient highlevel data structures and a simple but effective approach to objectoriented
More informationSeeVogh User Guide. Date: March 26 th, 2012
SeeVogh User Guide Date: March 26 th, 2012 About This Guide This guide provides users an overview of the features found in the SeeVogh PC client software. This guide does not discuss the use of the web
More informationGeoGebra. 10 lessons. Gerrit Stols
GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter
More informationMASKS & CHANNELS WORKING WITH MASKS AND CHANNELS
MASKS & CHANNELS WORKING WITH MASKS AND CHANNELS Masks let you isolate and protect parts of an image. When you create a mask from a selection, the area not selected is masked or protected from editing.
More informationMATLAB Workshop 14  Plotting Data in MATLAB
MATLAB: Workshop 14  Plotting Data in MATLAB page 1 MATLAB Workshop 14  Plotting Data in MATLAB Objectives: Learn the basics of displaying a data plot in MATLAB. MATLAB Features: graphics commands Command
More informationAdapted from Using PowerPoint To Create an Art History Presentation by Elizabeth S. Funk, NYU, Institute of Fine Arts
A Quick Start Guide to Using PowerPoint For Imagebased Presentations Knight Visual Resources Facility College of Architecture, Art and Planning Cornell University By Susan Jane Williams & William Staffeld
More informationMicrosoft Word defaults to left justified (aligned) paragraphs. This means that new lines automatically line up with the left margin.
Microsoft Word Part 2 Office 2007 Microsoft Word 2007 Part 2 Alignment Microsoft Word defaults to left justified (aligned) paragraphs. This means that new lines automatically line up with the left margin.
More informationGeneral Information Online Assessment Tutorial before Options for Completing the Online Assessment Tutorial
General Information Online Assessment Tutorial Schools must ensure every student participating in an online assessment has completed the Online Assessment Tutorial for the associated assessment at least
More informationMariemont City Schools
Mariemont City Schools Citrix Virtual Desktop Environment Citrix is a virtual desktop system that allows users to access their Mariemont Windows 7 desktop from anywhere with an Internet connection. Once
More informationThis training module reviews the CRM home page called the Dashboard including:  Dashboard My Activities tab.  Dashboard Pipeline tab
This training module reviews the CRM home page called the Dashboard including:  Dashboard My Activities tab  Dashboard Pipeline tab  My Meetings Dashlet  My Calls Dashlet  My Calendar Dashlet  My
More informationPiktochart 101 Create your first infographic in 15 minutes
Piktochart 101 Create your first infographic in 15 minutes TABLE OF CONTENTS 01 Getting Started 5 Steps to Creating Your First Infographic in 15 Minutes 1.1 Pick a Template 1.2 Click Create and Start Adding
More informationCREATING EXCEL PIVOT TABLES AND PIVOT CHARTS FOR LIBRARY QUESTIONNAIRE RESULTS
CREATING EXCEL PIVOT TABLES AND PIVOT CHARTS FOR LIBRARY QUESTIONNAIRE RESULTS An Excel Pivot Table is an interactive table that summarizes large amounts of data. It allows the user to view and manipulate
More informationSpreadsheets and Laboratory Data Analysis: Excel 2003 Version (Excel 2007 is only slightly different)
Spreadsheets and Laboratory Data Analysis: Excel 2003 Version (Excel 2007 is only slightly different) Spreadsheets are computer programs that allow the user to enter and manipulate numbers. They are capable
More informationSketchUp Instructions
SketchUp Instructions Every architect needs to know how to use SketchUp! SketchUp is free from Google just Google it and download to your computer. You can do just about anything with it, but it is especially
More informationPreparing a Slide Show for Presentation
In this chapter Find out why it s important to put finishing touches on a slide show Learn how to use the slide sorter Explore the use of slide transitions Learn how to change slide color schemes and backgrounds
More informationFireworks CS4 Tutorial Part 1: Intro
Fireworks CS4 Tutorial Part 1: Intro This Adobe Fireworks CS4 Tutorial will help you familiarize yourself with this image editing software and help you create a layout for a website. Fireworks CS4 is the
More informationGeneral Information Online Assessment Tutorial before Options for Completing the Online Assessment Tutorial
General Information Online Assessment Tutorial Schools must ensure every student participating in an online assessment has completed the Online Assessment Tutorial for the associated assessment at least
More informationWhat s New in SMART Board Software 9.0 for Windows Operating Systems
What s New in SMART Board Software 9.0 for Windows Operating Systems If you ve used SMART Board software before, you ll notice many changes in version 9.0. In this document, we ll introduce version 9.0
More informationChapter 4 Creating Charts and Graphs
Calc Guide Chapter 4 OpenOffice.org Copyright This document is Copyright 2006 by its contributors as listed in the section titled Authors. You can distribute it and/or modify it under the terms of either
More informationBeas Inventory location management. Version 23.10.2007
Beas Inventory location management Version 23.10.2007 1. INVENTORY LOCATION MANAGEMENT... 3 2. INTEGRATION... 4 2.1. INTEGRATION INTO SBO... 4 2.2. INTEGRATION INTO BE.AS... 4 3. ACTIVATION... 4 3.1. AUTOMATIC
More informationUSING MICROSOFT PHOTO EDITOR
USING MICROSOFT PHOTO EDITOR Create a new image 1 On the File menu, click New. 2 Select the Image type you want. 3 Enter the Resolution, Height, and Width you want. Open an existing image 1 On the Standard
More informationCONSTRUCTING SINGLESUBJECT REVERSAL DESIGN GRAPHS USING MICROSOFT WORD : A COMPREHENSIVE TUTORIAL
CONSTRUCTING SINGLESUBJECT REVERSAL DESIGN GRAPHS USING MICROSOFT WORD : A COMPREHENSIVE TUTORIAL PATRICK GREHAN ADELPHI UNIVERSITY DANIEL J. MORAN MIDAMERICAN PSYCHOLOGICAL INSTITUTE This document is
More informationTable of Contents ccconvert...1 verifycc...6 ccreview...8
1 ccconvert Table of Contents ccconvert...1 Controls and Displays...2 Main Interface Windows Version...2 Main Interface OSX Version...3 How to Use ccconvert...4 Setup...4 Connect Hardware...4 Install
More informationPrinting to the Poster Printer
Printing to the Poster Printer Document size The HP Design Jet Z3100ps uses a roll of paper that is 36 wide, however it does not print all the way to the edge of the paper (known as a bleed ). One dimension
More informationEXCEL EXERCISE AND ACCELERATION DUE TO GRAVITY
EXCEL EXERCISE AND ACCELERATION DUE TO GRAVITY Objective: To learn how to use the Excel spreadsheet to record your data, calculate values and make graphs. To analyze the data from the Acceleration Due
More informationWindows PowerShell Essentials
Windows PowerShell Essentials Windows PowerShell Essentials Edition 1.0. This ebook is provided for personal use only. Unauthorized use, reproduction and/or distribution strictly prohibited. All rights
More informationData Visualization. Brief Overview of ArcMap
Data Visualization Prepared by Francisco Olivera, Ph.D., P.E., Srikanth Koka and Lauren Walker Department of Civil Engineering September 13, 2006 Contents: Brief Overview of ArcMap Goals of the Exercise
More informationTutorial for Programming the LEGO MINDSTORMS NXT
Tutorial for Programming the LEGO MINDSTORMS NXT Table of contents 1 LEGO MINDSTORMS Overview 2 Hardware 2.1 The NXT Brick 2.2 The Servo Motors 2.3 The Sensors 3 Software 3.1 Starting a Program 3.2 The
More information